Aircraft: MD-11
Phase of flight: Cruise
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor; Inflight Event / Encounter Weather / Turbulence

I had run the APU on the ground; about 41-45 C ambient temperature; for air conditioning for about 45 minutes because the ground air cart was introducing noxious fumes into the aircraft. We started #3 engine in the blocks at Maintenance request; then pushed back and started #1. As #1 engine came on speed; received 'APU FAULT' and 'APU AUTO SHUTDOWN' messages. When APU de-energized; the alerts went away. We called on SATCOM and received a Maintenance [Deferral] for APU INOP. No attempt was made to restart APU since it was no longer needed for delayed engine start of #2 engine. Departed and received clearance to FL370 with Control. Approaching Waypoint ZZZZZ; entered cloud and selected Engine A/I (Anti-Ice) on. This checked with our route plot that clouds began just prior to ZZZZZ; but the limited amount of lightning shown was to the south of our track. Not too long after passing ZZZZZ with no presentation on weather radar of any rain or convective activity; we encountered severe turbulence lasting about a minute. The autopilot was unable to maintain altitude and we pitched down and were descending rapidly. The First Officer (Pilot Flying) disconnected the autopilot and did a very nice job minimizing altitude loss to about 400 ft. He stabilized the aircraft and put the autopilot back on. I notified [Center] via CPDLC (Controller-Pilot Datalink Communication) of a PIREP (Pilot Report) for severe turbulence; notified [Dispatcher] of the same; and broadcast on guard the turbulence report to other aircraft nearby. We were both puzzled that the weather radar had shown nothing. While in the turbulence; we had a lot of St. Elmo's fire on the windscreen and heard moderate rain on the fuselage. We suspected we had inadvertently entered convective activity. The weather radar tested properly during preflight and in flight it had widely scattered green dots; but no indication of weather. I varied the tilt and tried the other weather radar and nothing changed in the presentation to indicate any weather. We continued flight uneventfully for about another 15-20 minutes when the same thing happened and we entered moderate turbulence with occasional severe turbulence. At no time since selecting engine anti-ice on; had we received any 'ICE DETECTED' alert. At this point; I elected to descend to find smoother altitude first FL350; then continued to FL310. In this descent; we received IAS comparator alerts on our PFDs (Primary Flight Displays). [Requested priority handling] with ATC due to degraded aircraft handing and unreliable airspeed; turbulence; and the need to keep other aircraft away from us and to obtain assistance. We asked for ground speed readouts; weather avoidance information [as] we thought our weather radars were inoperative; and vectors while we sorted the situation out. Also selected EMG on ADS (Air Data Source) and advised [Dispatcher] of the emergency; our problem; and intentions. Captain's PFD was showing about 293 IAS; standby about 285-290; and First Officer's PFD approximately 278 knots IAS. I thought we might have encountered severe icing that impacted a pitot-static system; with no icing alert (because of the St. Elmo's on our windscreen once again and the sound of precipitation on the aircraft) and selected anti-ice all on as a precaution. We had a lot of alerts flashing on our displays; but the ones that seemed to be the most prevalent (that I could read in the turbulence) and remained the longest were level 2 alerts for 'SEL ELEV FEEL MAN' and 'SEL FLAP LIM OVRD' and level 1 alert 'WSHEAR DET FAIL'. Ran the QRH for unreliable airspeed. We suspected Captain's airspeed was correct; but couldn't conclusively determine that; so we followed the 'NO--able to identify unreliable Air Data Source?' path in the QRH as we established pitch and power settings and continued our descent to FL230. Didn't want to make a flyable situation worse by guessing the bad system when it wasn't conclusive and we had a flyable jet if either airspeed were correct. Once down at FL230 conditions improved considerably. The alerts on the display went away; so I put the AFS (Automatic Flight System) switches back up and both autopilot and auto throttles were available. ATC asked us if we wanted a full [priority handling]. I was unsure what that wording meant to the [local] authorities; but it seemed like a good idea because we still had odd indications in the aircraft and it was best to get the plane on the ground expeditiously. We were able to once again navigate; and proceeded direct to a RNAV point on the arrival and received priority handling for our arrival. The approach had been briefed up; and I informed the First Officer I would do the landing because of our situation and that I would keep a crosscheck of our Alpha and use the [instrument] information to do a visual landing in VMC conditions. I asked him to keep up with the pitch and power settings and if we had a recurrence of IAS miscompare we would continue using pitch and power and my HUD information. We elected to proceed to ZZZ1 for landing because it was VMC; we were already briefed for it; I was familiar with terrain and airfield; and it was on normal descent profile from our present position. I did not want to do an auto-land because we still had some questions about the jet. We received vectors to final; and were captured on the ILS with everything normal. We configured early in case of problems and I flew manually in case of problems with the automation. Even though I confirmed the FMAs (Flight Mode Annunciators) showed captured; our flight director and PFD guidance took us below glideslope with no correction. I corrected and transitioned to visual and cross checked with ILS raw data and disregarded the incorrect guidance information. Landing uneventfully; taxied to parking with fire trucks following. After debrief with Maintenance; called Duty Officer; Dispatcher; and Regional Director of Operations (at duty officer request). Local safety authorities request synopsis of events too.Unanticipated severe turbulence with no Pireps; Airmets; weather radar returns; etc. was the primary cause. Unknown if our equipment / instrument malfunctions were due to power interruptions; un-annunciated icing; or other cause. On the ground; I wrote up all the malfunctions and debriefed with Maintenance. They interrogated the systems and found faults in IRU (Inertial Reference Units) 1; IRU 2; IRU 3; ADC (Air Data Computers) 1; ADC 2; FCC (Flight Control Computers) 1; FCC 2; auto-land; and several other systems. APU shutdown history was for high oil temp. They also found numerous power interruption codes. I can't think of what I would have done differently given the same information. The weather conditions were unanticipated and weather radar didn't provide any returns of significance. We had a lot of instrument and equipment anomalies with no alerts besides the IAS miscompare and some other indications of unreliable airspeed. On landing; Maintenance found faults in all 3 IRUs; both ADCs; and both FCCs. Root cause of the lack of weather radar returns; icing detection system appearing not to work; and anomalies between ILS raw data; FMA displays; and flight director are unknown; but numerous power interruptions were recorded in fault lists --possibly due to the turbulence or affected by the APU auto shutdown on preflight? Or did we have severe icing which compromised a pitot static system and the lower altitude of FL230 help clear the anomalies? Local maintenance checked all systems; and the jet flew the following morning. The thing that still concerns me is the lack of alerting information for multiple systems that; to us; weren't working. We were at cruise at FL370 just passing point ZZZZZ. It was night and we were IMC with no significant weather returns on the weather radar. Without warning; we experienced severe turbulence for approximately one minute. Due to the turbulence; we lost approximately 400 ft. of altitude. I disengaged the autopilot and immediately began a climb back to our cleared altitude. Once back at cruise altitude; I reengaged the autopilot and we continued to experience light to moderate turbulence and then more severe turbulence. We informed [Center] of the severe turbulence and received clearance to descend in an effort to get out of the turbulence. We descended to FL350 as cleared. At FL350; the turbulence was occasional moderate but after a few minutes; we experienced an 'IAS Miscompare'. I continued flying and the Captain got into the QRH and executed the procedures. At this point; we also began receiving Level 2 'SEL ELEV FEEL MAN' and 'SEL FLAP LIM OVRD'. Along with Level 1 'WSHEAR DET FAIL' and 'NO ICE DETECTED'. At this point; we thought icing was causing our issues. The captain began running the appropriate QRH procedures and we asked for FL310 in an effort to leave the continued moderate turbulence and icing conditions. At this point I was hand flying as the QRH directed the AFS (Auto Flight System) shut off (IAS miscompare). As we approached FL310 with no relief from supposed icing conditions and unreliable airspeed; the Captain [requested priority handling] and requested lower in an effort to remove any icing from the aircraft. We had engine; wing; and windshield anti-ice on and but had a 'No Ice Detected' indication. We elected to land at ZZZ1 due to good weather and it was already loaded and briefed. After a few minutes at FL230; the IAS miscompare went away and we were able to reengage the flight director and autopilot. Captain; FO (First Officer); and standby airspeed indicators were consistent. Because we had [requested priority handling]; we had priority to land. We continued on vectors and executed an ILS to an uneventful landing. We then taxied to our gate. There are many possible combinations of causes of these events but the actual cause is unknown. It is unknown at this point if the weather radar was functioning properly and could have given us a warning of the impending area of convective activity. Later; the [Maintenance Technician] said there were faults on the ADCs (Air Data Computers); FCCs (Flight Control Computers); and the IRUs (Inertial Reference Units). There were also possible power transfer issues when we had an APU auto shutdown at ZZZ that may have resulted in electrical issues. Flight Crew reported experiencing APU problems during preflight; and indicated IAS miscompare; several instrument and equipment anomalies as well as severe turbulence in cruise flight.